To find the probability of being dealt exactly 4 aces in a 13-card hand from a standard 52-card deck, we can use the hypergeometric distribution. The total number of ways to choose 4 aces from 4 available is ( \binom{4}{4} = 1 ), and the number of ways to choose the remaining 9 cards from the 48 non-aces is ( \binom{48}{9} ). The total number of ways to choose any 13 cards from 52 is ( \binom{52}{13} ). Thus, the probability is given by ( \frac{1 \times \binom{48}{9}}{\binom{52}{13}} ).
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Since there are only four aces in a standard 52 card deck, the probability of being dealt five aces is zero.
There is a .53846 probability (53.846%) if aces are counted as face cards. If they are not counted the probability drops to .48077 (48.077%)
To get four Aces it is necessary to draw several cards. There is no information on how many cards are drawn. Whether or not they are drawn randomly.
The probability of drawing four aces in a draw of four cards from a deck of 52 cards is 3 in 812175. (4 in 52) times (3 in 51) times (2 in 50) times (1 in 49)
The probability of drawing two Aces from a standard deck of 52 cards is 4 in 52 times 3 in 51, or 12 in 2652, or 1 in 221, or about 0.00452.